is linear and invertible, essentially becuase of
Hodge theory and de Rham's theorem
(for a detailed argument, see [1]}).
Let \((\mathbf {a}, \, \mathbf {d})\in {A}^n\) . Suppose first that for any \(j\)  and \(k\) ,
the curve \(\gamma _k\)  does not contain the point \(a_j\) .
For any \(k\) , we choose an orthogonal tangent frame \(\lbrace \tau _{1,k}, \, \tau _{2,k}\rbrace \) 
defined in a neighbourhood of \(\gamma _k\) .
Let \(\mathcal {A}_k\)  be the connection 1-form induced
by \(\lbrace \tau _{1,k}, \, \tau _{2,k}\rbrace \) , as in (REF ). We define
\( \zeta _k(\mathbf {a}, \, \mathbf {d}) := \int _{\gamma _k} \left(\mathrm {d}^*\Psi (\mathbf {a}, \, \mathbf {d}) + \mathcal {A}_k\right)\qquad \textrm {for } k\in \lbrace 1, \, \ldots , \, 2\mathfrak {g}\rbrace ,\) 
